Let be the binary linear code
How many binary linear codes are there which are equivalent (under coordinate permutations) to ?
[THOUGHT] We need to determine how many distinct binary linear codes are equivalent to C under coordinate permutations. This means we count the number of distinct codes obtained by permuting the coordinates of C. The code C consists of all vectors where each pair of coordinates (1,2), (3,4), ..., (15,16) are equal. The stabilizer of C under the symmetric group on 16 coordinates consists of permutations that either preserve each pair or swap the two coordinates within a pair, and also permute the pairs among themselves. This stabilizer is isomorphic to (Z2)^8 ⋊ S8, so its size is 2^8 * 8!. The orbit size (number of distinct codes) is 16! divided by the stabilizer size. Computing this gives 2027025. [/THOUGHT] [PYTHON] import math
num = math.factorial(16) // (2**8 * math.factorial(8)) print(num) [/PYTHON]
[THOUGHT] The Python computation confirms that the number of distinct codes equivalent to C under coordinate permutations is 2027025. [/THOUGHT]